Extreme diffusion with point-sink killing field

Abstract

We study here the escape time for the fastest diffusing particle from the boundary of an interval with point-sink killing sources. Killing represents a degradation that leads to the probabilistic removal of the moving Brownian particles. We compute asymptotically the mean time it takes for the fastest particle escaping alive and obtain the extreme statistic distribution. These computations relies on an explicit expression for the time dependent flux of the Fokker-Planck equation using the time dependent Green's function and Duhamel's formula. We obtain a general formula for several point-sink killing, showing how they directly interact. The range of validity of the present formula for the mean extreme times of the fastest is evaluated with Brownian simulations. Finally, we discuss some applications to the early calcium signaling at neuronal synapses.